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Clever Proof That Math Has Its Charms
A Review by Paul A. Robinson Jr.

Is it possible to trisect an angle using only a straight edge and a compass? This is not the central point of Ian Stewart's Letters to a Young Mathematician, but it is one of the curious threads woven into these delightful letters written to "Meg," a fictitious mentee, as she advances through a career in mathematics.

Stewart begins by framing and discussing the questions that most non-mathematicians ask. "What is mathematics? Hasn't it all been done? What do mathematicians do?" If this was all this book did it would simply follow in the wake of G.H. Hardy's elegant 1940 memoir A Mathematician's Apology.

But fortunately, Stewart, who is a professor of mathematics at the University of Warwick in England, goes beyond that in these letters that follow Meg in roughly chronological order, from high school to a tenured position at a university.

Each chapter addresses one or two questions, beginning with "Why study mathematics?" and concluding with speculations on the nature of God. In the process, Stewart not only provides insight into the life of an academic, but also cleverly introduces the names of the greats of the math world, in addition to offering a booklist (from science fiction to biography) for the well read.

It's a brilliant way to help the reader develop fondness for the pursuit of mathematics without resorting to actual mathematical theorem and proof (although theorems are discussed as well).

Stewart hints, for instance, at the marvelous story of Wiles proving Fermat's last theorem. He tantalizes us with clever problems such as trisecting an angle. He evokes "the inner beauty of mathematics ... its ideas, the generalities, the sudden flashes of insight," but does so without taking the reader through the details.

To the mathematician, as Stewart explains, nature is one grand excuse for mathematics, "the development of mathematics is, and always has been, a two-way trade between real-world problems and symbolic or geometric methods devised to obtain answers. Of course math is effective for understanding nature; that, ultimately, is where it comes from."

So subjects like "bird crystals" - how birds arrange themselves on phone lines - and groupoids - "natural algebra's structure that replaces the symmetry group" - pop up in delightful ways. Stewart's illustrations are drawn from a variety of problems that demonstrate what mathematicians do and how they think, even as it taps into their excitement over both process and solution.

For nonmathematicians, Letters to a Young Mathematician offers wonderful insight into academics, a reading list in a variety of fields, and a bit of knowledge about Gauss, Fibonacci, Leibniz, Feynman, and Fermat. It also serves as a primer on mathematicians, their culture, their tribal customs, and their community. For mathematicians themselves, Stewart provides first-rate career advice and offers a charming example of how best to talk to the rest of us.

Paul A. Robinson Jr. is a professor of physics at Principia College